On the asymptotic normality of frequency polygons for strongly mixing spatial processes. Mohamed EL MACHKOURI

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1 On the asymptotic normality of frequency polygons for strongly mixing spatial processes Mohamed EL MACHKOURI Laboratoire de Mathématiques Raphaël Salem UMR CNRS 6085, Université de Rouen France Abstract This paper establishes the asymptotic normality of frequency polygons in the context of stationary strongly mixing random fields indexed by Z d. Our method allows us to consider only minimal conditions on the width bins and provides a simple criterion on the mixing coefficients. In particular, we improve in several directions a previous result by Carbon, Francq and Tran AMS Subject Classifications 2000: 62G05, 62G07, 60G60. Key words and phrases: Central limit theorem, spatial processes, random fields, nonparametric density estimation, frequency polygon, histogram, mixing. 1 Introduction and notations The frequency polygon is a density estimator based on the histogram. It has the advantage to be conceptually and computationaly simple since it just consists of linking the mid points of the histogram bars but its simplicity is not the only interest. In fact, for time series, Scott [16] as shown that the rate of convergence of frequency polygon is superior to the histogram for smooth densities. For other references on non-spatial density estimation based on the frequency polygon, one can refer to Beirlant et al. [1] and Carbon et al. [7]. To our knowledge, the only references in the spatial context are Carbon [5] and Carbon et al. [6] for strongly mixing random fields indexed by Z d and Bensaid and Dabo-Niang [2] for strongly mixing random fields indexed by R d. In [6] the asymptotic normality of the frequency polygon estimator is obtained under interleaved conditions on the width bin and the strong mixing coefficients. In this paper, we provide a simple criterion on the mixing coefficients for the frequency polygon to satisfy the central limit theorem when only minimal conditions on the width bin see Assumption A2 below are assumed. Our main result Theorem 1 improve Theorem 4.1 in [6] in several directions. Our approach which is based on the Lindeberg s method seems to

2 be superior to the so-called Bernstein s blocking method used in [6] but also in many others papers on nonparametric estimation for random fields see [2], [8], [9], [12], [17]. For another application of the Lindeberg s method, one can refer to El Machkouri [11] where the asymptotic normality of the Parzen-Rosenblatt kernel density estimator is proved for strongly mixing random fields improving a previous result by Tran [17] obtained also by the Bernstein s blocking method an coupling arguments. Note that the central limit theorem in [11] is obtained for random fields observed on squares Λ n of Z d but actually the result still holds if the regions Λ n are only assumed to have cardinality going to infinity as n goes to infinity. In particular, it is not neccessary to impose any condition on the boundary of Λ n. Let d be a positive integer and let X i i Z d be a stationary real random field defined on some probability space Ω, F, P with an unknown marginal density f. For any finite subset B of Z d, denote B the number of elements in B. In the sequel, we assume that we observe X i i Z d on a sequence Λ n n 1 of finite subsets of Z d such that n goes to infinity as n goes to infinity. We lay emphasis on that we do not impose any condition on the boundary of the regions Λ n. Given two σ-algebras U and V, we recall the α-mixing coefficient introduced by Rosenblatt [15] and defined by αu, V = sup{ PA B PAPB, A U, B V} and the ρ-mixing coefficient introduced by Kolmogorov and Rozanov [13] and defined by { } Covf, g ρu, V = sup, f L 2 U, g L 2 V. f 2 g 2 It is well known that 4αU, V ρu, V see [4]. For any τ in N { } and any positive integer n, we consider the mixing coefficients α 1,τ n and ρ 1,τ n defined by α 1,τ n = sup {ασx k, F B, k Z d, B τ, ΞB, {0} n}, ρ 1,τ n = sup {ρσx k, F B, k Z d, B τ, ΞB, {0} n} where F B = σx i ; i B for any subset B of Z d and the distance Ξ is defined for any subsets B 1 and B 2 of Z d by ΞB 1, B 2 = min{ i j, i B 1, j B 2 } where i j = max 1 k d i k j k for any i = i 1,..., i d and j = j 1,..., j d in Z d. We say that the random field X i i Z d lim n ρ 1,τ n = 0 for some τ in N { } respectively. is α-mixing or ρ-mixing if lim n α 1,τ n = 0 or Let b n n 1 be a sequence of positive real numbers going to zero when n goes to infinity. For each n in N, we consider the partition I n,k k Z of the real line defined for any k in 2

3 Z by I n,k = [k 1b n, kb n. Let n, k be fixed in N Z and let I n,k and I n,k+1 be two adjacent histogram bins. Denote ν n,k and ν n,k+1 the numbers of observations falling in these intervals respectively. The values of the histogram in these previous bins are given by ν n,k n b n 1 and ν n,k+1 n b n 1 and the frequency polygon f n,k is defined for x J n,k := [ k 1 2 bn, k bn by f n,k x = Define Y i,s = 11 Xi I n,s k x νn,k 1 + b n n b n 2 k + x νn,k+1. b n n b n for any s in Z, then f n,k x = 1 n b n i Λ n a k xy i,k + a k xy i,k+1 where a s u = 1 +s u 2 b n and a s u = a s u for any s in Z and any u in J n,s. Finally, we consider the normalized frequency polygon estimator f n defined for any x in R such that fx > 0 by f n x = k Z 2 f n,k x σ n,k x 11 J n,k x where σn,kx 2 1 = k xbn fx. Our main results are stated in Section 2 and the proofs are given in Section 3. 2 Main results We consider the following assumptions. A1 The density function f is differentiable and its derivative f is locally bounded. A2 b n goes to zero such that n b n goes to infinity as n goes to infinity. A3 sup j 0 P X 0 I n,s, X j I n,t = Ob 2 n for any s and t in Z. B3 P X 0 I n,s, X j I n,t = ob n for any s, t and j in Z. Remark 1. Obviously B3 is weaker than A3. Moreover, if the joint density f 0,j of X 0, X j exists then A3 is true by assuming that sup j 0 f 0,j is locally bounded whereas B3 is true by assuming only that f 0,j is locally bounded for each j 0. As in Theorem 3.1 in [6], the following result gives the asymptotic variance of f n. 3

4 Proposition 1 Assume that A1 and A2 hold. If one of the following assumptions i A3 holds and m 1 m2d 1 α 1,1 m <, ii B3 holds and m 1 md 1 ρ 1,1 m <, is true then lim n n b n Vf n x = 1 for any x in R such that fx > 0. Our main result is the following central limit theorem. Theorem 1 Assume that A1 and A2 hold. If one of the following assumptions i A3 holds and m 1 m2d 1 α 1, m <, ii B3 holds and m 1 md 1 ρ 1, m <, is true then for any positive integer r and any distinct points x 1,..., x r in R such that fx i > 0 for any 1 i r, f n x 1 Ef n x 1 n b n 1/2. f n x r Ef n x r where Id is the unit matrix of order r. Law N 0, Id 1 n Remark 2. Theorem 1 improves Theorem 4.1 in [6] in three directions: the regions Λ n where the random field is observed are not reduced to rectangular ones we do not assume any boundary condition, the assumption A2 on the width bin b n is minimal and the α-mixing condition is weaker than the one assumed in Theorem 4.1 in [6], that is α 1, m = Om θ with θ > 2d. 3 Proofs Throughout this section, the symbol κ will denote a generic positive constant which the value is not important and we recall that i = max 1 k d i k for any i = i 1,..., i d Z d. Let τ N { } be fixed and consider the sequence m n,τ n 1 defined by 1 m n,τ = max v n, 1 d i d α 1,τ i + 1 b n i >v n where v n = [ b 1 ] 2d n and [. ] denotes the integer part function. The following technical lemma is a spatial version of a result by Bosq, Merlevède and Peligrad [3], pages

5 Lemma 1 Let τ N { } be fixed. If m 1 m2d 1 α 1,τ m < then m n,τ, m d n,τ b n 0 and 1 i d α m d 1,τ i 0. n,τ b n i >m n,τ Proof of Lemma 1. First, m n,τ goes to infinity since b n goes to zero and m n,τ v n. We consider the function ψ defined for any m in N by ψm = i >m i d α 1,τ i. Since m 1 m2d 1 α 1,τ m <, we have ψm converges to zero as m goes to infinity. { bn ψ } Consequently, m d n,τb n max, κ vn + b n 0. Moreover, noting n that m d n,τ 1 b n ψ vn 1 b n ψ mn,τ since v n m n,τ, we derive also 1 m d n,τb n The proof of Lemma 1 is complete. i >m n,τ i d α 1,τ i ψm n,τ n 0. Proof of Proposition 1. For any n 1 and any x in R, n b n Vf n x = k Z n b n Vf n,k x σ 2 n,k x 11 Jn,k x. Let x in R such that fx > 0. For any integer n 1, we denote by kn the unique integer such that x belongs to J n,kn. It suffices to show that n b n Vf n,kn x lim n σn,kn 2 x = 1. In the sequel, we write k instead of kn and we denote p s = I n,s fudu for any s in Z. We have n b n Vf n,k x = a 2 kx p k1 p k + 1 CovY i,k, Y j,k b n n b n i,j Λn i j + a 2 kx p k+11 p k CovY i,k+1, Y j,k+1 b n n b n i,j Λn i j + 2a k xa k x p kp k CovY i,k, Y j,k+1. b n n b n i,j Λn i j 5

6 Denote also w n,k x = 1 b n a 2 k xp k 1 p k + a 2 kxp k+1 1 p k+1 2a k xa k xp k p k+1. Arguing as in the proof of Lemma 3.2 in [6], we have by Taylor expansion p k = b n fx + b n 2 2kb n x b n f c k p k+1 = b n fx + b n 2 2kb n x + b n f c k+1 where c k J n,k and c k+1 J n,k+1. Then for j = k or j = k + 1, Consequently max{0, fxb n κb 2 n} p j fxb n + κb 2 n. 3 max{0, b n f 2 x 2b 2 nκfx + κ 2 b 3 n} p kp k+1 b n b n f 2 x + 2b 2 nκfx κ 2 b 3 n 4 and for j = k or j = k + 1, max{0, fx κ + f 2 xb n + κ 2 b 3 n} p j1 p j b n fx + κ f 2 xb n + κ 2 b 3 n. 5 Finally, we obtain lim n w n,k x = 1. σn,k 2 x Let s, t be equal to k, k, k, k + 1 or k + 1, k + 1. It suffices to show that lim n 1 n b n i,j Λn i j CovY i,s, Y j,t = 0 6 By stationarity of the random field X i i Z d, we have 1 CovY i,s, Y j,t 1 CovY 0,s, Y j,t. 7 n b n b i,j Λn n i j Using 3, for j 0, we have CovY 0,s, Y j,t Y 0,s 2 Y 0,t 2 ρ 1,1 j = p s p t ρ 1,1 j κb n ρ 1,1 j. 8 Moreover, using again 3, for any j 0, j Z d j 0 CovY 0,s, Y j,t P X 0 I n,s, X j I n,t + κb 2 n. 9 6

7 Assuming B3 and m 1 md 1 ρ 1,1 m < and combining 8 and 9 with the dominated convergence theorem, we obtain CovY 0,s, Y j,t = o1. 10 b n j Z d j 0 Finally, 6 follows from inequality 7. Similarly, by Rio s covariance inequality cf. [14], Theorem 1.1, CovY 0,s, Y j,t 2 2ασY0,s,σY j,t 0 Q Y0,s uq Yj,t udu where Q Z u = inf{t; P Z > t u} for any u in [0, 1]. bounded by 1, we derive Using 9 and assuming A3, we derive Since Y 0,s and Y j,t are CovY 0,s, Y j,t 4α 1,1 j. 11 sup CovY 0,s, Y j,t κb 2 n. 12 j 0 Assuming m 1 m2d 1 α 1,1 m < and combining 11, 12 and Lemma 1, we obtain CovY 0,s, Y j,t κ m d b n,1b n + 1 j d α n m d 1,1 j = o1 13 n,1b n j >m n,1 j Z d j 0 where m n,1 n 1 is defined by 2. Finally, using inequality 7, we obtain again 6. The proof of Proposition 1 is complete. Remark 3. The reader should note that the asymptotic variance given in Theorem 3.1 in [6] is not the good one. In fact, using the notations in [6], it should be 1/2 + 2k 0 x/b 2 fx instead of 1/2 + 2k 0 x/b 2 fx. Proof of Theorem 1. Without loss of generality, we assume that r = 2 and we denote x and y in place of x 1 and x 2. Let λ 1 and λ 2 be fixed in R such that λ λ 2 2 = 1. For any i in Z d, we define i = λ 1 Z i x + λ 2 Z i y where for any u in R such that fu > 0, Z i u = 1 bn s Z a s uy i,s EY i,s + a s uy i,s+1 EY i,s+1 11 Jn,s u, σ n,s u s u 2 b n fu. a s u = s u b n, a s u = a s u and σ 2 n,su = 7

8 Lemma 2 E and n i W E 0 i κ W b n + o1 for any finite subset W of Z d \{0}. Proof of Lemma 2. Recall that p s = I n,s fudu and J n,s = [s 1 2 b n, s b n for any s in Z. We have and b n EZ 0 xz 0 y = E 2 0 = λ 2 1EZ 2 0x + λ 2 2EZ 2 0y + 2λ 1 λ 2 EZ 0 xz 0 y 14 k,l Z 2 + k,l Z 2 + k,l Z 2 + k,l Z 2 a k xa l y σ n,k xσ n,l y EY 0,k p k Y 0,l p l 11 Jn,k J n,l x, y a k xa l y σ n,k xσ n,l y EY 0,k p k Y 0,l+1 p l+1 11 Jn,k J n,l x, y a k xa l y σ n,k xσ n,l y EY 0,k+1 p k+1 Y 0,l p l 11 Jn,k J n,l x, y a k xa l y σ n,k xσ n,l y EY 0,k+1 p k+1 Y 0,l+1 p l+1 11 Jn,k J n,l x, y For n sufficiently large, x, y belongs to J n,k J n,l with k l 2. Then for any s = k or s = k + 1 and t = l or t = l + 1, EY 0,s p s Y 0,t p t = p s p t κb 2 n. Since 0 a s u 1, 0 a s u 1 and σ 2 n,su fu/2 > 0 for any u in J n,s such that fu > 0 and any s in Z, we obtain Similarly, for any u in R, we have b n EZ 2 0u = k Z + 2 k Z + k Z EZ 0 xz 0 y κb n. 15 a 2 k u σ 2 n,k uey 0,k p k 2 11 Jn,k u a k ua k u σn,k 2 u EY 0,k p k Y 0,k+1 p k+1 11 Jn,k u a 2 k u σ 2 n,k uey 0,k+1 p k Jn,k u. Noting that EY 0,s p s 2 = p s 1 p s and EY 0,s p s Y 0,s+1 p s+1 = p s p s+1 for any s in Z and keeping in mind 4 and 5, we obtain for any u in R, EZ 2 0u n

9 Combining 14, 15 and 16, we obtain E 2 0 n λ2 1 + λ 2 2 = 1. Let W be a finite subset of Z d \{0} and let i W be fixed. We have E 0 i λ 2 1E Z 0 xz i x + 2 λ 1 λ 2 E Z 0 xz i y + λ 2 2E Z 0 yz i y. 17 If u and v are fixed in R then b n E Z 0 uz i v k,l Z 2 + k,l Z 2 + k,l Z 2 + k,l Z 2 Noting that for any s and t in Z, we derive a k ua l v σ n,k uσ n,l v E Y0,k p k Y i,l p l 11Jn,k J n,l u, v a k ua l v σ n,k uσ n,l v E Y 0,k p k Y i,l+1 p l+1 11Jn,k J n,l u, v a k ua l v σ n,k uσ n,l v E Y0,k+1 p k+1 Y i,l p l 11Jn,k J n,l u, v a k ua l v σ n,k uσ n,l v E Y0,k+1 p k+1 Y i,l+1 p l+1 11Jn,k J n,l u, v. E Y 0,s p s Y i,t p t EY 0,s Y i,t + 3p s p t CovY 0,s, Y i,t + κb 2 n, E Z 0 uz i v k,l Z 2 + k,l Z 2 + k,l Z 2 + k,l Z 2 + κb n. a k ua l v CovY 0,k, Y i,l 11 Jn,k J σ n,k uσ n,l v b n,l u, v n a k ua l v CovY 0,k, Y i,l+1 11 Jn,k J σ n,k uσ n,l v b n,l u, v n a k ua l v CovY 0,k+1, Y i,l 11 Jn,k J σ n,k uσ n,l v b n,l u, v n a k ua l v CovY 0,k+1, Y i,l+1 11 Jn,k J σ n,k uσ n,l v b n,l u, v n Assuming B3 and m>0 md 1 ρ 1,1 m < and using 10 or assuming A3 and m>0 m2d 1 α 1,1 m < and using 13, we obtain E Z 0 uz i v κ W b n + o1. 18 i W 9

10 Combining 17 and 18, we obtain i W E 0 i κ W b n + o1. The proof of Lemma 2 is complete. We are going to follow the Lindeberg-type proof of the central limit theorem for stationary random fields established in [10]. Let ϕ be a one to one map from [1, κ] N to a finite subset of Z d and ξ i i Z d denote S ϕk ξ = a real random field. For all integers k in [1, κ], we k ξ ϕi and Sϕkξ c = i=1 with the convention S ϕ0 ξ = S c ϕκ+1 ξ = 0. On the lattice Zd we define the lexicographic order as follows: if i = i 1,..., i d and j = j 1,..., j d are distinct elements of Z d, the notation i < lex j means that either i 1 < j 1 or for some p in {2, 3,..., d}, i p < j p and i q = j q for 1 q < p. To describe the set Λ n, we define the one to one map ϕ from [1, n ] N to Λ n by: ϕ is the unique function such that ϕk < lex ϕl for 1 k < l n. From now on, we consider a field ξ i i Z d of i.i.d. random variables independent of X i i Z d such that ξ 0 has the standard normal law N 0, 1. We introduce the fields Γ and γ defined for any i in Z d by Γ i = Let h be any function from R to R. i n 1/2 and γ i = ξ i n 1/2 κ i=k ξ ϕi For 0 k l n + 1, we introduce h k,l Γ = hs ϕk Γ+S c ϕl γ. With the above convention we have that h k, n +1Γ = hs ϕk Γ and also h 0,l Γ = hs c ϕl γ. In the sequel, we will often write h k,l instead of h k,l Γ. We denote by B 4 1R the unit ball of C 4 b R: h belongs to B4 1R if and only if h is bounded by 1, belongs to C 4 R and its first four derivatives are also bounded by 1. It suffices to prove that for all h in B 4 1R, We use Lindeberg s decomposition: E h S ϕn Γ n E h ξ 0. E h S ϕn Γ h ξ 0 n = E h k,k+1 h k 1,k. Now, h k,k+1 h k 1,k = h k,k+1 h k 1,k+1 + h k 1,k+1 h k 1,k. 10

11 Applying Taylor s formula we get that: and h k,k+1 h k 1,k+1 = Γ ϕk h k 1,k Γ2 ϕkh k 1,k+1 + R k h k 1,k+1 h k 1,k = γ ϕk h k 1,k γ2 ϕkh k 1,k+1 + r k where R k Γ 2 ϕk 1 Γ ϕk and r k γ 2 ϕk 1 γ ϕk. Since Γ, ξ i i ϕk is independent of ξ ϕk, it follows that h E γ ϕk h k 1,k+1 = 0 and E γϕkh 2 k 1,k+1 k 1,k+1 = E n Hence, we obtain E hs ϕn Γ h ξ 0 n = EΓ ϕk h k 1,k+1 n + E Γ 2 ϕk 1 h k 1,k+1 n 2 n + E R k + r k. Let 1 k n be fixed. Since 0 is bounded by κ/ b n, applying Lemma 2 we derive Consequently, we obtain E R k E 0 3 n 3/2 κ n 3 b n 1/2 and E r k E ξ 0 3 n 3/2. n E R k + r k = O Now, it is sufficient to show EΓ ϕk h k 1,k+1 + E n lim n 1 n b n + 1 1/2 n 1/2 = o1. Γ 2 ϕk 1 h k 1,k+1 = n 2 First, we focus on n E Γ ϕk h k 1,k+1. Let the sets {V k i ; i Z d, k N } be defined as follows: Vi 1 = {j Z d ; j < lex i} and Vi k = Vi 1 {j Z d ; i j k} for k 2. Let N n n 1 be a sequence of positive integers satisfying N n such that N d nb n

12 For all n in N and all integer 1 k n, we define E n k = ϕ[1, k] N V Nn ϕk and S n ϕk Γ = i E n k For any function Ψ from R to R, we define Ψ n k 1,l = ΨS n ϕk Γ + Sc ϕl γ. We are going to apply this notation to the successive derivatives of the function h. Our aim is to show that n lim n E Γ ϕk h k 1,k+1 Γ ϕk S ϕk 1 Γ S n ϕk Γ h k 1,k+1 = First, we use the decomposition Γ ϕk h k 1,k+1 = Γ ϕk h n k 1,k+1 + Γ ϕk Γ i. h k 1,k+1 h n k 1,k+1. We consider a one to one map ψ from [1, E n k ] N to E n k such that ψi ϕk ψi 1 ϕk. For any subset B of Z d, recall that F B = σx i ; i B and set E M X i = EX i F V M i, M N, i Z d. The choice of the map ψ ensures that S ψi Γ and S ψi 1 Γ are F ψi ϕk V -measurable. ϕk The fact that γ is independent of Γ imply that E Γ ϕk h S cϕk+1 γ = 0. Therefore E E Γ ϕk h n k 1,k+1 = n k E Γ ϕk θ i θ i 1 22 i=1 where θ i = h S ψi Γ + Sϕk+1. c γ Since S ψi Γ and S ψi 1 Γ are F ψi ϕk V - measurable, we can take the conditional expectation of Γ ϕk with respect to F V ψi ϕk ϕk in the right hand side of 22. On the other hand the function h is 1-Lipschitz, hence θ i θ i 1 Γ ψi. Consequently, E Γϕk θ i θ i 1 E Γψi E ψi ϕk Γϕk and Hence, n E n k E Γ ϕk h n k 1,k+1 i=1 k 1,k+1 E Γ ϕk h n 1 n n E Γ ψi E ψi ϕk Γ ϕk. E n k E ψi E ψi ϕk ϕk i=1 j N n j E j 0 1. ϕk 12

13 For any j in Z d, we have j E j 0 1 = Cov j 11 E j E j 0 <0, 0 and consequently j E j ρ 1, j. By Lemma 2, we know that 0 2 is bounded. So, assuming m 0 md 1 ρ 1, m <, we derive n k 1,k+1 E Γ ϕk h n = o1. 23 By Rio s covariance inequality cf. [14], Theorem 1.1, we have also j E j α1, j 0 Q 2 0 udu where Q 0 is defined by Q 0 u = inf{t 0 ; P 0 > t u} for any u in [0, 1]. Since 0 is bounded by κ/ b n, we have Q 0 κ/ b n and j E j 0 1 κ b n α 1, j. Assuming m 0 m2d 1 α 1, m < and choosing N n = m n, recall that m n, n 1 is defined by 2 and satisfies m n, such that m d n, b n 0, we derive n k 1,k+1 E Γ ϕk h n κ m d n, b n j m n, j d α 1, j. By Lemma 1, we obtain again 23. Now, applying Taylor s formula, Γ ϕk h k 1,k+1 h n k 1,k+1 = Γ ϕks ϕk 1 Γ S n ϕk Γh k 1,k+1 + R k, where R k 2 Γ ϕks ϕk 1 Γ S n ϕk Γ1 S ϕk 1Γ S n ϕk Γ. Consequently n 21 holds if and only if lim n E R k = 0. In fact, denoting W n = { N n + 1,..., N n 1} d and Wn = W n \{0}, we have n 1 E R k 2E 0 i 1 n 1/2 i i W n i W n = 2E i 1 1 i Wn n 1/2 i i W n 2 E 2 n 0 1/2 i + 2 E 0 i. i W n i W n 13

14 Since 0 κ bn, we derive n κ E R k E n b n 1/2 0 i + 2 E 0 i i W n i Wn κ E 2 0 n b n + κ 1/2 n b n + 2 E 1/2 0 i i Wn 1 = O n b n + 1 N 1/2 n b n + 2 d nb 1/2 n + o1 by Lemma 2 = o1 by 20 and Assumption A2. In order to obtain 19 it remains to control n F 0 = E h k 1,k+1 Γ 2 ϕk 2 + Γ ϕk S ϕk 1 Γ S n ϕk Γ 1. 2 n Applying Lemma 2, we have F 0 E 1 n h n k 1,k+1 2 ϕk E E E 0 j j V0 1 Wn E 1 n h n k 1,k+1 2 ϕk E ON nb d n + o1. Since N d nb n 0, it suffices to prove that F 1 = E 1 n h n k 1,k+1 2 ϕk E 2 0 goes to zero as n goes to infinity. In fact, keeping in mind W n = { N n + 1,..., N n 1} d and W n = W n \{0}, we have F 1 1 n L 1,k + L 2,k n 14

15 where L 1,k = E h n k 1,k+1 2 ϕk E 2 0 = 0 since h n k 1,k+1 is F V mn, -measurable ϕk and L 2,k = E h k 1,k+1 h n 2 k 1,k+1 ϕk E 0 2 E 2 i 2 n 1/2 0 i W n E 2 κ 0 n b n + i W E 0 n i since 1/2 n b n 1/2 0 κ a.s. bn 1 = O n b n + N nb d n + o1 by Lemma 2 1/2 n b n 1/2 = o1 by 20 and Assumption A2. The proof of Theorem 1 is complete. Acknowledgments. reading and constructive comments. The author thanks an anonymous referee for his\her careful References [1] J. Beirlant, A. Berlinet, and L. Györfi. On piecewise linear density estimators. Statist. Neerlandica, 533: , [2] N. Bensaïd and S. Dabo-Niang. Frequency polygons for continuous random fields. Stat. Inference Stoch. Process., 131:55 80, [3] D. Bosq, Merlevède F., and M. Peligrad. Asymptotic normality for density kernel estimators in discrete and continuous time. J. Multivariate Anal., 68:78 95, [4] R. C. Bradley. Basic properties of strong mixing conditions. A survey and some open questions. Probab. Surv., 2: , [5] M. Carbon. Polygone des fréquences pour des champs aléatoires. C. R. Math. Acad. Sci. Paris, 3429: , [6] M. Carbon, C. Francq, and L.T. Tran. Asymptotic normality of frequency polygons for random fields. Journal of Statistical Planning and Inference, 140: ,

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